Find the slope of the line joining the points (3, -4) and (5, 2).
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Understanding the Slope of a Line
The slope of a line is a measure of its steepness. It tells us how much the vertical position (y-coordinate) changes for every unit change in the horizontal position (x-coordinate). The slope is often denoted by the letter 'm'.
Formula for Calculating Slope
To find the slope of a straight line passing through two distinct points, say \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\), we use the following formula:
\(\displaystyle m = \frac{\text{Change in y}}{\text{Change in x}} = \frac{y_2 - y_1}{x_2 - x_1}\)
Applying the Slope Formula to the Given Points
We are asked to find the slope of the line joining the points (3, -4) and (5, 2).
Let's identify our points and their coordinates:
| Point | x-coordinate | y-coordinate |
|---|---|---|
| Point 1 (\(P_1\)) | \(x_1 = 3\) | \(y_1 = -4\) |
| Point 2 (\(P_2\)) | \(x_2 = 5\) | \(y_2 = 2\) |
Now, we substitute these coordinates into the slope formula:
\(\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}\)
\(\displaystyle m = \frac{2 - (-4)}{5 - 3}\)
First, calculate the difference in the y-coordinates (\(y_2 - y_1\)):
\(2 - (-4) = 2 + 4 = 6\)
Next, calculate the difference in the x-coordinates (\(x_2 - x_1\)):
\(5 - 3 = 2\)
Now, divide the change in y by the change in x:
\(\displaystyle m = \frac{6}{2}\)
\(\displaystyle m = 3\)
Resulting Slope Calculation
The slope of the line joining the points (3, -4) and (5, 2) is 3.
Revision Table: Slope Calculation Key Points
| Concept | Description |
|---|---|
| Slope Definition | Measure of line steepness; \(\frac{\text{Change in y}}{\text{Change in x}}\) |
| Slope Formula | \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for points \((x_1, y_1)\) and \((x_2, y_2)\) |
| Given Points | (3, -4) and (5, 2) |
| Calculated Slope | 3 |
Additional Information on Linear Equations
Understanding the slope is fundamental in coordinate geometry and linear equations. Here are some related concepts:
- Types of Slopes: A line can have a positive slope (rises from left to right), a negative slope (falls from left to right), a zero slope (horizontal line), or an undefined slope (vertical line).
- Equation of a Line: The slope-intercept form of a linear equation is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept (the point where the line crosses the y-axis).
- Parallel Lines: Two distinct non-vertical lines are parallel if and only if they have the same slope.
- Perpendicular Lines: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. A vertical line and a horizontal line are also perpendicular.
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Important Questions from Co-ordinate Geometry
In which ratio the point (-3, p) divides the line segment joining the points (-5, -4) and (-2, 3)?
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In which quadrant both abscissa and ordinate are negative?
Find the value of K for which equation x – Ky = 2, 3x + 2y = 5 has unique solution.
The area of a triangle ABC, whose coordinates are A(x 1, y 1), B(x 2, y 2), C(x 3, y 3) is given by______.
